Did you man y = sqrt (x)?
The moment of inertia is
I = ∫r²dm
In this case, the integration is along the x-axis from 0 to X (y is not defined for x < 0). The moment must be limited to finite distance, since it will go to ∞ if x --> ∞
dm = ρds, where ρ is the linear density.
r = y = √x; r² = x
ds = √[1 + y'²]dx where y' = dy/dx = (1/2)x^-(1/2)
ds = √[1 + (1/4)x^(-1)]dx
Then
I =ρ∫x√[1 + 1/(4x)]dx
I =ρ/2∫√[4x² + x)]dx
ANALYZING THE SOLUTION MIGHT JUST HELP YOU TO ANSWER THE PROBLEM ON YOUR OWN!